#### Date of Original Version

10-2007

#### Type

Article

#### Abstract or Description

We consider the problem of learning mixtures of product distributions over discrete domains in the distribution learning framework introduced by Kearns et al. [Proceedings of the 26th Annual Symposium on Theory of Computing (STOC), Montr´eal, QC, 1994, ACM, New York, pp. 273–282]. We give a poly(n/ε)-time algorithm for learning a mixture of k arbitrary product distributions over the n-dimensional Boolean cube {0, 1}n to accuracy ε, for any constant k. Previous polynomial-time algorithms could achieve this only for k = 2 product distributions; our result answers an open question stated independently in [M. Cryan, Learning and Approximation Algorithms for Problems Motivated by Evolutionary Trees, Ph.D. thesis, University of Warwick, Warwick, UK, 1999] and [Y. Freund and Y. Mansour, Proceedings of the 12th Annual Conference on Computational Learning Theory, 1999, pp. 183–192]. We further give evidence that no polynomial-time algorithm can succeed when k is superconstant, by reduction from a difficult open problem in PAC (probably approximately correct) learning. Finally, we generalize our poly(n/ε)-time algorithm to learn any mixture of k = O(1) product distributions over {0, 1, . . . , b − 1}n, for any b = O(1).

#### DOI

10.1137/060670705

#### Published In

SIAM Journal of Computing, 37, 5, 1536-1564.